Computer Generated Islamic Star Patterns
Craig S. Kaplan
Department of Computer Science and Engineering
University of Washington
Box 352350, Seattle, WA 98195-2350 USA
csk@cs.washington.edu
Abstract
Islamic star patterns are a beautiful and highly geometric art form whose original design techniques
are lost in history. We describe one procedure for constructing them based on placing radially-symmetric
motifs in a formation dictated by a tiling of the plane, and show some styles in which they can be
rendered. We also show some results generated with a software implementation of the technique.
1 Introduction
More than a thousand years ago, Islamic artisans began to adorn
architectural surfaces with geometric patterns. As the centuries
passed, this practice developed into a rich system of intricate
ornamentation that followed the spread of Islamic culture into
Africa, Europe, and Asia. The ornaments often took the form of
a division of the plane into star-shaped regions, which we will
simply call “Islamic star patterns”; a typical example appears
on the right. To this day, architectural landmarks in places like
Granada, Spain and Isfahan, Iran demonstrate the artistic mastery achieved by these ancient artisans.
Lurking in these geometric wonders is a long-standing
historical puzzle. The original designers of these figures kept
their techniques a closely guarded secret. Other than the finished
works themselves, little information survives about the thought
process behind their star patterns.
Many attempts have been made to reinvent the design process for star patterns, resulting in a variety
of successful analyses and constructions. Grünbaum and Shephard [9] decompose periodic Islamic patterns
by their symmetry groups, obtaining a fundamental region they use to derive properties of the original
pattern. Abas and Salman apply this decomposition process to a large collection of patterns [2]. Elsewhere,
they argue for a simple approach tied to the tools available to designers of the time [1]. Dewdney proposes
a method of reflecting lines off of periodically-placed circles [5]. Castera presents a technique based on the
construction of networks of eightfold stars and “safts” [7].
This paper presents a technique described by Hankin [10], based on his experiences seeing partiallyfinished installations of Islamic art. It also incorporates the work of Lee [11], who provides simple constructions for the common features of Islamic patterns. Given a tiling of the plane containing regular polygons
and irregular regions, we fill the polygons with Lee’s stars and rosettes, and infer geometry for the remaining regions. We have implemented this technique as a Java applet, which was used to produce the
examples in this paper. The applet is available for experimentation at http://www.cs.washington.
edu/homes/csk/taprats/.
The rest of the paper is organized as follows. Section 2 presents constructions for the common
features of Islamic patterns: stars and rosettes. Section 3 shows how complete designs may be built using
repeated copies of those features. Techniques for creating visually appealing renderings of the designs are
given in Section 4. Some results appear in Section 5. The paper concludes in Section 6 by exploring some
opportunities for future work.
2
Stars and Rosettes
In our method, a regular n-gon is filled with a figure of symmetry type dn (which has all the symmetries of
the n-gon). In practice, these figures belong to a small number of families which we describe below.
For n ≥ 3, let the unit circle be parameterized via γ(t) = (cos (2πt/n), sin (2πt/n)). We construct
the n-pointed star polygon (n/d) by drawing, for 0 ≤ i < n, the line segment σi connecting γ(i) and
γ(i + d). Note that d < n/2 and that (n/1) is the regular n-gon. For some values of k 6= i, σi will intersect
σk , dividing σi into a number of subsegments. We often choose to draw only the first s subsegments at either
end of σi , which we indicate with the extended notation (n/d)s. Figure 1 shows the different stars that are
possible when n = 8.
Our implementation generalizes this construction, allowing d to take on any real value in [1, n/2).
When d is not an integer, point P is computed as the intersection of line segments γ(i)γ(i + d) and
γ(i + bdc − d)γ(i + bdc), and σi is replaced by the two line segments γ(i)P and P γ(i + bdc). Two examples of this generalization are given in Figure 2.
γ(i-0.6)
γ(i)
(8/1)1
(8/2)1
(8/2)2
(8/3)1
(8/3)2
(8/3)3
P
γ(i+3)
γ(i+3.6)
(8/3.6)2
Figure 1 The six possible eight-pointed stars when
d is an integer.
Figure 2 An (n/d)s star for non-integral d.
When sixfold stars are arranged as on the left side of Figure 3, a higher-level structure emerges:
every star is surrounded by a ring of regular hexagons. The pattern can be regarded as being composed of
these surrounded stars, or rosettes. Placing copies of the rosette in the plane will leave behind gaps, which
in this case happen to be more sixfold stars.
The rosette, a central star surrounded by hexagons, appears frequently in Islamic art. They do not
only appear in the sixfold variety, meaning that we must generalize the construction of the rosette to handle
A
C
E
D
B
F
O
Figure 3 An arrangement of sixfold stars can be
reinterpreted as rosettes. The pattern is one of the
oldest in the Islamic tradition.
Figure 4 The construction of a ten-pointed rosette.
arbitrary n. The construction given by Lee [11] yields an n-fold rosette for any n ≥ 5 while preserving
most of the symmetry of the hexagons. Each hexagon has four edges not adjacent to the central star; all four
edges are congruent. Moreover, the outermost edges lie on the regular n-gon joining the rosette’s tips, and
the two “radial” edges are parallel.
A diagram of Lee’s construction process is shown in Figure 4. To begin, inscribe a regular n-gon
in the unit circle and draw the n-gon whose vertices bisect its edges. Let A and B be adjacent vertices
of the outer n-gon and C and D be adjacent vertices of the inner n-gon with D bisecting AB. The key
is to then identify point E, computed as the intersection of CD with the bisector of 6 OAB. Then F is
the intersection of OA with the line through E parallel to OD. The rest of the rosette follows through
application of symmetry group dn : edges DE and EF lead to the outer edges of the hexagons, while copies
of F become the points of the inner star, which can be completed with the construction given earlier.
By sliding E along the bisector of 6 OAB, we can continuously vary the shape of the rosette while
preserving the congruence of the four outer hexagonal edges.
Some Islamic designs feature a motif slightly more complicated than a basic
rosette, where opposing limiting edges from adjacent tips of the rosette are joined up.
The resulting object has the same symmetries and number of outer points as the rosette,
but with an additional layer of geometry on its outside. We refer to these as “extended
rosettes”. A ninefold extended rosette appears on the right.
3 Filling the Plane
Equipped with a taxonomy of typically Islamic motifs that can be inscribed in regular polygons, we are now
ready to create complete periodic designs. We start with a periodic tiling containing regular polygons, with
irregular polygons thrown in as needed to fill gaps. For each regular n-gon with n > 4, we choose an n-fold
star, rosette or extended rosette to place in it and replicate that motif everywhere the n-gon appears in the
tiling. The motif is placed so that its points bisect the edges of the n-gon.
The result is a design like that of Figure 5(b). There are still large gaps where motifs were not
placed, corresponding here to to the squares in the original tiling. Each square edge is adjacent to an edge
of an octagon, and so a vertex of the chosen motif is incident to it. The presence of these vertices suggests a
technique for filling the gaps in a natural way, by extending the line segments that terminate on the boundary
of the region until they meet other extended segments in the region’s interior. Except for degenerate cases,
(a)
(b)
(c)
(d)
Figure 5 Given the octagon and square tiling shown in (a), we decide to place 8-fold rosettes in the
octagons and let the system infer geometry for the squares. The rosette is copied to all octagons in (b), and
lines from unattached tips are extended into the interstitial spaces until they meet in (c). The construction
lines are removed, resulting in the final design shown in (d).
Figure 6 Some alternative patterns based on the octagon-square tiling that can be constructed by varying
the motif placed in the octagons.
following this procedure guarantees that the resulting design will admit an interlacing.
Figure 5(c) shows the design with the free rosette tips extended into the gaps. Here, the natural
extension creates regular octagons in the interstitial regions. To complete the construction, the original
tiling is removed, resulting in the design in Figure 5(d), a well-known Islamic star pattern [3, plate 48].
Given a tiling containing regular polygons and gaps, we can now construct a wide range of different
designs by choosing different motifs for the regular polygons. Even when restricted to the octagon-square
tiling used above, many different designs can be created. Three alternative designs appear in Figure 6. Of
course, we can expand the range of this technique in the other dimension by also encoding a large number
of different tilings.
The implementation currently encodes fourteen tilings from which Islamic star patterns may be produced. Some are familiar regular or semi-regular tilings [8, Section 2.1]. Some are derived by examination
of well-known Islamic patterns. The remaining tilings were discovered by experimentation and lead to novel
Islamic designs shown in Section 5.
4
Rendering
The output of the construction process is a planar graph. To be sure, the graph has an intrinsic beauty that
holds up when it is rendered as simple line art. Historically, however, these designs were never merely
drawn as lines. Islamic star patterns are typically used as a decoration for walls and floors. The faces of
the planar graph are realized as a mosaic of small terracotta tiles in a style known as “Zellij”. Often, the
edges are thickened and incorporated into the mosaic with narrow tiles, sometimes broken up to suggest an
interlacing pattern. Islamic designs can also be found carved into wood or stone and built into trellises and
latticework.
Plain
Outline
Emboss
Interlace
Checkerboard
Outline and
Checkerboard
Figure 7 Rendering styles.
To increase the aesthetic appeal of our implementation, we provide the ability to render the planar
graph in a manner reminiscent of some of these techniques (see Figure 7). The outline style thickens the
edges of the planar graph, adding weight and character to the lines of the plain style. The emboss style adds a
3D effect to the outline style, simulating the appearance of a wooden trellis; the centre of each thickened edge
is raised towards the viewer and the graph is rendered by specifying the direction of a fictitious light source.
The interlace style adds line segments at each crossing to suggest an over-under relationship between the
crossing edges. When every vertex in the graph has degree two or four, the crossings can always be chosen so
that the graph is broken into strands that adhere to a strict alternation of over and under in their intersections
with other strands. The final style, checkerboard, renders the faces of the graph and not the edges. When
all vertices have even degree (as they must in an interlace design), it is always possible to colour the faces
with only two colours in such a way that faces with the same colour never share an edge. The checkerboard
style walks the graph, creating a consistent 2-colouring.
A further enhancement can be achieved by layering one of the edge-based rendering styles on top of
the checkerboard style. This combination comes closest to the appearance of Zellij.
5
Results
Figures 8 and 9 present a selection of finished computer-generated drawings. The first group, Figure 8, is
made up of reproductions of well-known Islamic star patterns which can be found in Bourgoin [3] or Abas
and Salman [2]. Figure 9 contains designs that do not appear in either of those sources. Three of them
are based on polygonal tilings that do not seem to be used by any known designs. These last three are
moderately successful, though they seem to lack the harmonious balance of the well-known designs. Still,
in an artform with a thousand-year tradition, any sort of novel design is certainly of interest.
6
Future Work
Our software implementation and the technique on which it is based allow access to a wide variety of designs
without offering so much flexibility that it becomes overly easy to wander out of the space of recognizably
Islamic patterns. There are, however, opportunities for future work that do not compromise the focus of the
system.
The set of available tilings from which to form patterns is open-ended. More tilings could be implemented. Some new ones can easily be derived by inspection of patterns in Bourgoin or Abas and Salman.
We could move away from periodicity by implementing aperiodic tilings with regular polygons. Castera
has constructed several ingenious aperiodic Islamic star pattern based on Penrose rhombs [4]. Finally, the
hyperbolic plane offers tremendous freedom in the construction of tilings with regular polygons. We hope
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
Figure 8 Some sample results based on well-known tilings from Islamic ornament. Each final design is
based on the corresponding tiling in the top row.
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
Figure 9 Sample results not found in the literature. The pattern in (a) is similar to one found in Abas and
Salman [2, p. 93], using extended rosettes instead of ordinary rosettes. The other three patterns are based
on previously unused tilings.
Figure 10 A novel pattern with 7-stars. The design in the centre, resulting from the natural extension
of star edges, leaves behind disproportionate octagons. The design on the right, constructed manually,
corrects this by redistributing the area to new 5-stars.
to adapt the technique described in this paper to the Poincaré model of the hyperbolic plane, much the same
way Dunham has done with Escher patterns [6].
One last aspect of the system we hope to improve is the naı̈ve extension of lines into interstitial
regions. Our algorithm can easily fail to produce attractive results. In Figure 10, a novel grid based on regular
heptagons is turned into an Islamic pattern by placing (7/3)2 stars in the heptagons. The natural extension
of star edges into the gaps leaves large, unattractive octagonal areas. With the appropriate heuristics, our
inference algorithm could detect cases such as this and add some complexity to the inferred geometry in
order to improve the final design.
Acknowledgments
I am indebted to Mamoun Sakkal for his guidance and helpful discussion while initiating this project in his
course. Thanks also to David Salesin for his valuable input and to Reza Sarhangi for the encouragement.
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